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This paper presents some analytical results on production and order dynamics in the context of a discrete-time VMI supply chain system composed of one retailer and one manufacturer. We firstly derive the lower bound and upper bound on the range of inventory fluctuations for the retailer under unknown demand. We prove that the production fluctuations can be interestingly smoothed and stabilized independent of the delivery frequency of the manufacturer used to satisfy the retailer’s demand, even if the retailer subsystem is unstable. The sufficient and necessary stability condition for the whole supply chain system is obtained. To further explore the production fluctuation problem, the bullwhip effect under unknown demand is explored based on a transfer function model with the purpose of disclosing the influences of parameters on production fluctuations. Finally, simulation experiments are used to validate the theoretical results with respect to inventory and production fluctuations.

Vendor-managed inventory (VMI) is a well-known collaborative program [

The benefits of VMI for both suppliers and retailers have been extensively studied in the literature [

Most of the existing literature has focused on game theory or optimization models [

In a VMI system, the upstream supplier has the flexibility to determine the amount and timing of the replenishment order for the retailer. However, as mentioned, the supplier might be penalized once the retailer’s inventory exceeds the predetermined range. Therefore, in order to reduce penalty cost, the supplier should know how the replenishment parameters for the retailer affect the range of inventory fluctuations of the retailer. In addition, the supplier should also consider the production or ordering cost and inventory cost for the whole system when designing its own replenishment policies. In this paper, we aim to answer three questions for the implementation of VMI:

Specifically, we attempt to explore the inventory and production fluctuations in the context of a VMI supply chain system composed of a manufacturer and a retailer. The manufacturer uses a reorder point policy to manage the retailer’s inventory and the APIOBPCS (automatic pipeline, inventory, and order based production control system) for production control [

The remainder of this paper is organized as follows. Section

Over the past decades, the advantages of VMI program have been extensively discussed. The shift of the authority of inventory control from downstream enables the supplier to choose the time and quantity for each replenishment order and also the route to transport the required goods to multiple retailers. It was well recognized that VMI brings more advantages over information sharing alone [

Although the opportunities of VMI are evident in many industries, the benefit allocation and coordination problem has been argued continuously [

In addition to the previous literature, optimal replenishment policies of VMI supply chain systems with multiple retailers have been studied frequently over the recent years [

Consider a periodic-review, single item VMI supply chain system composed of a retailer and a manufacturer. In a VMI program, the retailer should provide the manufacturer with demand and inventory information. On the basis of these information, the manufacturer makes replenishment decision to control the inventory for both the manufacturer and the retailer.

The timing of the events in each period is assumed as follows. At the beginning of each period

To be more general, we assume that the demand of the retailer is unknown but bounded by

In a VMI system, a vendor makes replenishment decisions based on information sharing with respect to downstream member’s demand and inventory. In our periodic inventory model, the balanced equations for the retailer’s inventory level,

The exponential smoothing algorithm is accurate for short-term forecasting and easy to implement [

The replenishment rule used by the manufacturer for production decision is called APIOBPCS [

In the literature, the production smoothing behavior is usually characterized by bullwhip effect metrics [

From the replenishment rule (

From (

A state space model can be developed directly from (

Before we discuss the dynamics of the VMI system, we shall firstly introduce the definition of stability. By stability, we mean that the output of a system is bounded if the input is bounded. The customer demand

Although the stability of the VMI system has been discussed in Disney and Towill [

In this section, we begin by studying the stability and inventory oscillation of the retailer. In the VMI program, the retailer may assume the risk of incurring high inventory cost or poor customer service level. To avoid such a problem, the retailer can increase its benefits by signing effective contractual agreements, which are characterized by the penalty cost once the retailer’s inventory beyond an upper bound or a lower bound. To reduce the penalty cost and maintaining satisfactory service level for the retailer, the manufacturer must clarify how the replenishment rule affects the range of inventory fluctuations of the retailer. To this end, as a first step, we shall now discuss the inventory fluctuation range of the retailer under uncertain demand with Theorem

Assume that

Let

The schematic dynamic curves for the retailer’s inventory position and reorder point.

From Theorem

Under the VMI program, the manufacturer manages both its own inventory and the retailer’s inventory based on the systematic information. The shifting of the authority of inventory control and information sharing, two main properties of a VMI supply chain system, must affect the dynamics properties of the entire system. Disney and Towill [

As mentioned, if

In fact, the stability of (

Let

where

It should be remarked that

In order to simplify the selection of replenishment parameters ensuring the system is stable, we can further obtain the stability condition independent of the production lead time, which is introduced in Theorem

The VMI supply chain is stable if

In fact, the precise stable regions gradually converge to the stable region independent of production lead times as the production lead time increases. Figure

The stability boundaries for the VMI supply chain system.

From above, we see that the production process of the manufacturer can be smoothed even with low delivery frequency determined by the manufacturer to control retailer’s inventory. In this sense, the manufacturer can reduce transportation cost and production cost simultaneously. However, the manufacturer must be cautious to determine the replenishment frequency or the size of order batching, because there still exists a trade-off between inventory cost and production cost caused by production fluctuations. This trade-off will be further explored via simulation experiments in the next section.

In the simulation experiments, we will firstly validate the theoretical results on the range of inventory fluctuations of the retailer subsystem. Then, we will study the dynamics interaction between the retailer and the manufacturer. Finally, the impact of different parameters on the production smoothing will be numerically studied with the worst-case bullwhip measure.

We will validate the theoretical results with uniformly distributed demand

In the simulations, we compute the retailer’s system performance by four metrics: replenishment frequency,

The other two metrics

The comparison between simulation results and theoretical results for the retailer subsystem.

| | | | |||||||||
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| | | | | | | | | | | | |

| | | 4 | 1.5 | 2 | -160 | 200 | 360 | 0.57 | 0.99 | 128.30 | -27.11 |

| | 4 | 1.5 | 2 | -360 | 420 | 780 | 0.52 | 0.97 | 265.67 | -63.48 | |

| | 4 | 1.5 | 2 | -560 | 640 | 1200 | 0.50 | 0.95 | 372.84 | -125.67 | |

| ||||||||||||

| 10 | 40 | | 1.5 | 2 | -180 | 120 | 300 | 0.58 | 0.43 | -60.96 | 64.32 |

10 | 40 | | 1.5 | 2 | -100 | 260 | 360 | 0.52 | 1.00 | 185.89 | 7.28 | |

10 | 40 | | 1.5 | 2 | -180 | 300 | 480 | 0.54 | 1.00 | 298.74 | 98.37 | |

| ||||||||||||

| 10 | 40 | 6 | | 2 | -100 | 260 | 360 | 1.00 | 0.48 | 44.30 | -40.93 |

10 | 40 | 6 | | 2 | -180 | 300 | 480 | 0.41 | 1.00 | 199.87 | 19.29 | |

10 | 40 | 6 | | 2 | -420 | 420 | 840 | 0.16 | 1.00 | 307.03 | 17.53 | |

| ||||||||||||

| 10 | 40 | 4 | 1.5 | | -220 | 200 | 420 | 0.43 | 0.81 | 101.57 | -53.32 |

10 | 40 | 4 | 1.5 | | -340 | 200 | 540 | 0.41 | 0.16 | 72.70 | -108.38 | |

10 | 40 | 4 | 1.5 | | -460 | 200 | 660 | 0.565 | 0.01 | 22.90 | -173.32 |

Table

We shall now simulate the system with a unit step signal as the demand input, which is represented by

In the case when

The inventory and production dynamics for the VMI supply chain system.

Production and order dynamics when

Inventory dynamics when

Production and order dynamics when

Inventory dynamics when

The advantages of

Here we simply test the stability results for

Validation of stability results.

For a manufacturer, reducing production fluctuations is significant because it is closely related to production cost. Smoothing production process helps arranging labor force, deciding the manufacturing capacity, and also enlarging the life of production machines. It is well recognized that production fluctuations can be smoothed by mitigating the bullwhip effect [

The bullwhip effect problem has received considerable attention since it causes high cost for upstream firms in a supply chain firm due to the distortion of demand information. The majority of the literature focuses on the bullwhip effect of supply chain system for specific demand with statistical method and control theory method [

Assume that

Using Matlab 7.0, we can calculate the value of

When

The impact of system parameters on the worst-case bullwhip.

This paper has investigated the dynamics of a VMI supply chain system composed of one manufacturer and one retailer. To provide guidelines for the manufacturer to better manage the retailer’s inventory, we firstly explored the fluctuation range of the retailer’s inventory. We found that the retailer’s inventory fluctuations are mainly influenced by demand characteristics, delivery frequency, lead time, and other parameters. Specifically, the magnitude of the retailer’s inventory fluctuation is an increasing function of the parameter

The shifting of the authority to manage the retailer’s inventory and information sharing certainly affect the dynamics for the manufacturer. Based on the difference equations, we investigated the dynamical properties of the entire VMI supply chain system. It is interesting to demonstrate that, under the VMI program, the production process can be stabilized even if the retailer subsystem is unstable. This result seems to be favorable for the manufacturer; however, the retailer will face serious shortages because the manufacturer fails to satisfy the demand of the retailer. Until present, the majority of the literature on the dynamics of the VMI supply chain system is limited to simulation studies. For example, Disney et al. [

Based on the difference equations, we also studied the stability of the VMI supply chain. Stability is a fundamental problem for production smoothing. Although the production decision of the manufacturer is based on system inventory, we found that the stability condition of the VMI supply chain is very similar to a one-echelon production and inventory control system, as has been studied in Wei et al. [

It is worth noting that this paper focuses on a VMI supply chain with a relatively simple structure and specific assumptions. In practice, the VMI program can be implemented for multiple supplier systems or distribution networks. In addition, a realistic supply chain system must be nonlinear [

The paper is focused on modeling and simulation. All the data is generated by simulation experiments with different parameter settings or available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper

This work was supported by the National Natural Science Foundation of China (nos. 71401181 and 71701213) and the MOE (Ministry of Education in China) Project of Humanities and Social Sciences (nos. 14YJC630136 and 15YJC630008).